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Metamath Proof Explorer

Theorem cdleme3e

Description: Part of proof of Lemma E in Crawley p. 113. Lemma leading to cdleme3fa and cdleme3 . (Contributed by NM, 6-Jun-2012)

Ref Expression
Hypotheses cdleme1.l 
|- .<_ = ( le ` K )
cdleme1.j 
|- .\/ = ( join ` K )
cdleme1.m 
|- ./\ = ( meet ` K )
cdleme1.a 
|- A = ( Atoms ` K )
cdleme1.h 
|- H = ( LHyp ` K )
cdleme1.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdleme1.f 
|- F = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) )
cdleme3.3 
|- V = ( ( P .\/ R ) ./\ W )
Assertion cdleme3e 
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ ( P .\/ Q ) ) ) ) -> V e. A )

Proof

Step Hyp Ref Expression
1 cdleme1.l 
 |-  .<_ = ( le ` K )
2 cdleme1.j 
 |-  .\/ = ( join ` K )
3 cdleme1.m 
 |-  ./\ = ( meet ` K )
4 cdleme1.a 
 |-  A = ( Atoms ` K )
5 cdleme1.h 
 |-  H = ( LHyp ` K )
6 cdleme1.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
7 cdleme1.f 
 |-  F = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) )
8 cdleme3.3 
 |-  V = ( ( P .\/ R ) ./\ W )
9 simpl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ ( P .\/ Q ) ) ) ) -> ( K e. HL /\ W e. H ) )
10 simpr1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ ( P .\/ Q ) ) ) ) -> ( P e. A /\ -. P .<_ W ) )
11 simpr3l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ ( P .\/ Q ) ) ) ) -> R e. A )
12 hllat 
 |-  ( K e. HL -> K e. Lat )
13 12 ad2antrr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ ( P .\/ Q ) ) ) ) -> K e. Lat )
14 eqid 
 |-  ( Base ` K ) = ( Base ` K )
15 14 4 atbase 
 |-  ( R e. A -> R e. ( Base ` K ) )
16 11 15 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ ( P .\/ Q ) ) ) ) -> R e. ( Base ` K ) )
17 simpr1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ ( P .\/ Q ) ) ) ) -> P e. A )
18 14 4 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
19 17 18 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ ( P .\/ Q ) ) ) ) -> P e. ( Base ` K ) )
20 simpr2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ ( P .\/ Q ) ) ) ) -> Q e. A )
21 14 4 atbase 
 |-  ( Q e. A -> Q e. ( Base ` K ) )
22 20 21 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ ( P .\/ Q ) ) ) ) -> Q e. ( Base ` K ) )
23 simpr3r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ ( P .\/ Q ) ) ) ) -> -. R .<_ ( P .\/ Q ) )
24 14 1 2 latnlej1l 
 |-  ( ( K e. Lat /\ ( R e. ( Base ` K ) /\ P e. ( Base ` K ) /\ Q e. ( Base ` K ) ) /\ -. R .<_ ( P .\/ Q ) ) -> R =/= P )
25 13 16 19 22 23 24 syl131anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ ( P .\/ Q ) ) ) ) -> R =/= P )
26 25 necomd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ ( P .\/ Q ) ) ) ) -> P =/= R )
27 1 2 3 4 5 lhpat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R e. A /\ P =/= R ) ) -> ( ( P .\/ R ) ./\ W ) e. A )
28 9 10 11 26 27 syl112anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ ( P .\/ Q ) ) ) ) -> ( ( P .\/ R ) ./\ W ) e. A )
29 8 28 eqeltrid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ ( P .\/ Q ) ) ) ) -> V e. A )